Riesz External Field Problems on the Hypersphere and Optimal Point Separation

We consider the minimal energy problem on the unit sphere 𝕊d in the Euclidean space ℝd+1 in the presence of an external field Q, where the energy arises from the Riesz potential 1/r s (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d − 2 ≤ s < d − 1. The proof uses a maximum principle for measures supported on 𝕊d. When Q is the Riesz s-potential of a signed measure and d − 2 ≤ s < d, our results lead to explicit point-separation estimates for (Q,s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on 𝕊d with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.